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A projective description of weighted inductive limits. (English) Zbl 0599.46026
Considering countable locally convex inductive limits of weighted spaces of continuous functions, if \({\mathcal V}=\{V_ n\}_ n\) is a decreasing sequence of systems of weights on a locally compact Hausdorff space X, we prove that the topology of \({\mathcal V}_ 0C(X)=_{n\to}C(V_ n)_ 0(X)\) can always be described by an associated system \(\bar V=\bar V_{{\mathcal V}}\) of weights on X; the continuous seminorms on \({\mathcal V}_ 0C(X)\) are characterized as weighted supremum norms. If \({\mathcal V}=\{v_ n\}_ n\) is a sequence of continuous weights on X, a condition is derived in terms of \({\mathcal V}\) which is both necessary and sufficient for the completeness (respectively, regularity) of the (LB)-space \({\mathcal V}_ 0C(X)\), and which is also equivalent to \({\mathcal V}_ 0C(X)\) agreeing algebraically and topologically with the associated weighted space \(C\bar V_ 0(X)\); for sequence spaces, this condition is the same as requiring that the corresponding echelon space be quasinormable.
A number of consequences follow. As our main application, in the case of weighted inductive limits of holomorphic functions, we obtain, using purely functional-analytic methods, a considerable extension of a theorem due to B. A. Taylor [Duke Math. J. 38, 379-385 (1971; Zbl 0214.378)], which is useful in connection with analytically uniform spaces and convolution equations. The projective description of weighted inductive limits also serves to improve upon existing tensor and slice product representations. Most of our work is in the context of spaces of scalar or Banach space valued functions, but, additionally some results for spaces of functions with range in certain (LB)-spaces are mentioned.

MSC:
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
30H05 Spaces of bounded analytic functions of one complex variable
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[1] Albert Baernstein II, Representation of holomorphic functions by boundary integrals, Trans. Amer. Math. Soc. 160 (1971), 27 – 37. · Zbl 0225.30044
[2] Carlos A. Berenstein and Milos A. Dostal, Analytically uniform spaces and their applications to convolution equations, Lecture Notes in Mathematics, Vol. 256, Springer-Verlag, Berlin-New York, 1972. · Zbl 0237.47025
[3] Klaus-Dieter Bierstedt, Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. I, J. Reine Angew. Math. 259 (1973), 186 – 210 (German). · Zbl 0252.46039
[4] Klaus-Dieter Bierstedt, Injektive Tensorprodukte und Slice-Produkte gewichteter Räume stetiger Funktionen, J. Reine Angew. Math. 266 (1974), 121 – 131 (German). · Zbl 0277.46019
[5] Klaus-D. Bierstedt, The approximation property for weighted function spaces, Function spaces and dense approximation (Proc. Conf., Univ. Bonn, Bonn, 1974), Inst. Angew. Math., Univ. Bonn, Bonn, 1975, pp. 3 – 25. Bonn. Math. Schriften, No. 81. Klaus-D. Bierstedt, Tensor products of weighted spaces, Function spaces and dense approximation (Proc. Conf., Univ. Bonn, Bonn, 1974), Inst. Angew. Math., Univ. Bonn, Bonn, 1975, pp. 26 – 58. Bonn. Math. Schriften, No. 81.
[6] Klaus-Dieter Bierstedt and Reinhold Meise, Bemerkungen über die Approximations-eigenschaft lokalkonvexer Funktionenräume, Math. Ann. 209 (1974), 99 – 107 (German). · Zbl 0267.46015
[7] Klaus-Dieter Bierstedt and Reinhold Meise, Induktive Limites gewichteter Räume stetiger und holomorpher Funktionen, J. Reine Angew. Math. 282 (1976), 186 – 220 (German). · Zbl 0318.46034
[8] K.-D. Bierstedt, B. Gramsch, and R. Meise, Lokalkonvexe Garben und gewichtete induktive Limites \?-morpher Funktionen, Function spaces and dense approximation (Proc. Conf., Univ. Bonn, Bonn, 1974), Inst. Angew. Math., Univ. Bonn, Bonn, 1975, pp. 59 – 72. Bonn. Math. Schriften, No. 81 (German). · Zbl 0335.46018
[9] Seán Dineen, Holomorphic functions on strong duals of Fréchet-Montel spaces, Infinite dimensional holomorphy and applications (Proc. Internat. Sympos., Univ. Estadual de Campinas, S ao Paulo, 1975) North-Holland, Amsterdam, 1977, pp. 147 – 166. North-Holland Math. Studies, Vol. 12, Notas de Mat., No. 54.
[10] Seán Dineen, Holomorphic germs on compact subsets of locally convex spaces, Functional analysis, holomorphy, and approximation theory (Proc. Sem., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1978) Lecture Notes in Math., vol. 843, Springer, Berlin, 1981, pp. 247 – 263.
[11] Leon Ehrenpreis, Fourier analysis in several complex variables, Pure and Applied Mathematics, Vol. XVII, Wiley-Interscience Publishers A Division of John Wiley & Sons, New York-London-Sydney, 1970. · Zbl 0195.10401
[12] Jean Pierre Ferrier, Spectral theory and complex analysis, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematics Studies, No. 4, Notas de Matemática (49). · Zbl 0261.46051
[13] Klaus Floret, Lokalkonvexe Sequenzen mit kompakten Abbildungen, J. Reine Angew. Math. 247 (1971), 155 – 195 (German). · Zbl 0209.43001
[14] Alain Goullet de Rugy, Espaces de fonctions pondérables, Israel J. Math. 12 (1972), 147 – 160 (French, with English summary). · Zbl 0238.46024
[15] Alexandre Grothendieck, Sur les espaces (\?) et (\?\?), Summa Brasil. Math. 3 (1954), 57 – 123 (French). · Zbl 0058.09803
[16] Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955), 140 (French). · Zbl 0064.35501
[17] O. v. Grudzinski, Convolutions-Gleichungen in Räumen von Beurling-Distributionen endlicher Ordnung, Habilitationsschrift, Kiel, 1980.
[18] Sönke Hansen, On the ”fundamental principle” of L. Ehrenpreis, Partial differential equations (Warsaw, 1978) Banach Center Publ., vol. 10, PWN, Warsaw, 1983, pp. 185 – 201.
[19] Sönke Hansen, Localizable analytically uniform spaces and the fundamental principle, Trans. Amer. Math. Soc. 264 (1981), no. 1, 235 – 250. · Zbl 0482.46023
[20] Ralf Hollstein, Inductive limits and \?-tensor products, J. Reine Angew. Math. 319 (1980), 38 – 62. · Zbl 0426.46053
[21] J. Horváth, Topological vector spaces and distributions. I, Addison-Wesley, Reading, Mass., 1966.
[22] Gert Kleinstück, Duals of weighted spaces of continuous functions, Function spaces and dense approximation (Proc. Conf., Univ. Bonn, Bonn, 1974), Inst. Angew. Math., Univ. Bonn, Bonn, 1975, pp. 98 – 114. Bonn. Math. Schriften, No. 81. · Zbl 0345.46023
[23] Gottfried Köthe, Topological vector spaces. I, Translated from the German by D. J. H. Garling. Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York Inc., New York, 1969. · Zbl 0179.17001
[24] Reinhold Meise, Räume holomorpher Vektorfunktionen mit Wachstumsbedingungen und topologische Tensorprodukte, Math. Ann. 199 (1972), 293 – 312 (German). · Zbl 0233.46050
[25] Hermann Neus, Über die Regularitätsbegriffe induktiver lokalkonvexer Sequenzen, Manuscripta Math. 25 (1978), no. 2, 135 – 145 (German, with English summary). · Zbl 0389.46058
[26] Bent E. Petersen, Holomorphic functions with growth conditions, Trans. Amer. Math. Soc. 206 (1975), 395 – 406. · Zbl 0321.32001
[27] João Prolla, The approximation property for Nachbin spaces, Approximation theory and functional analysis (Proc. Internat. Sympos. Approximation Theory, Univ. Estadual de Campinas, Campinas, 1977) North-Holland Math. Stud., vol. 35, North-Holland, Amsterdam-New York, 1979, pp. 371 – 382.
[28] D. A. Raĭkov, A criterion of completeness of locally convex spaces, Uspehi Mat. Nauk 14 (1959), no. 1 (85), 223 – 229 (Russian).
[29] Jean Schmets, Espaces de fonctions continues, Lecture Notes in Mathematics, Vol. 519, Springer-Verlag, Berlin-New York, 1976 (French). · Zbl 0334.46022
[30] Laurent Schwartz, Espaces de fonctions différentiables à valeurs vectorielles, J. Analyse Math. 4 (1954/55), 88 – 148 (French). · Zbl 0066.09601
[31] Laurent Schwartz, Théorie des distributions à valeurs vectorielles. I, Ann. Inst. Fourier, Grenoble 7 (1957), 1 – 141 (French). · Zbl 0089.09601
[32] Claude Servien, Sur la topologie d’un espace de fonctions entières avec poids, Séminaire Pierre Lelong (Analyse), Année 1974/75, Springer, Berlin, 1976, pp. 90 – 95. Lecture Notes in Math., Vol. 524 (French).
[33] W. H. Summers, Weighted locally convex spaces of continuous functions, Ph. D. Dissertation, Louisiana State University, Baton Rouge, 1968.
[34] W. H. Summers, A representation theorem for biequicontinuous completed tensor products of weighted spaces, Trans. Amer. Math. Soc. 146 (1969), 121 – 131. · Zbl 0189.12702
[35] W. H. Summers, Dual spaces of weighted spaces, Trans. Amer. Math. Soc. 151 (1970), 323 – 333. · Zbl 0203.12401
[36] B. A. Taylor, The fields of quotients of some rings of entire functions, Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966) Amer. Math. Soc., Providence, R.I., 1968, pp. 468 – 474.
[37] B. A. Taylor, A seminorm topology for some (\?\?)-spaces of entire functions, Duke Math. J. 38 (1971), 379 – 385. · Zbl 0214.37802
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